Subgroups — #14647517

Order 2816.

422 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13068732	C₂	1	2	yes	maximal
2	C₄×C₈×Dic₁₁	C₂	1	1	yes	maximal, characteristic
3	C₂×#10622096	C₂	1	1	yes	maximal, characteristic
4	C₂×#10622118	C₂	1	2	yes	maximal
5	C₂×C₁₁×#335	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₁×C₄⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₁₁	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₁₁	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₁₁	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₁₁	C₄	1	1	yes	characteristic
12	C₄²×Dic₁₁	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	2	yes	—
14	C₂×C₄×C₁₁⋊C₈	C₂²	1	1	yes	characteristic
15	C₁₁×#335	—	2	2	no	—
16	#10622188	—	2	4	no	—
17	#10622118	—	2	4	no	—
18	#10622096	—	2	2	no	—
19	C₂²×C₈×C₁₁	—	2	2	no	—
20	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
23	C₂×C₄×Dic₁₁	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×Dic₁₁	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×Dic₁₁	D₄	1	2	yes	—
26	C₂×C₄×Dic₁₁	Q₈	1	2	yes	—
27	C₂²×C₁₁⋊C₈	—	2	2	no	—
28	C₂²×C₁₁⋊C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₁₁⋊C₈	C₂×C₄	1	1	yes	characteristic
30	C₁₁×C₄⋊C₈	—	2	2	no	—
31	C₁₁×C₄⋊C₈	—	2	2	no	—
32	C₄×C₈×C₁₁	D₄	1	2	yes	—
33	C₄×C₈×C₁₁	Q₈	1	2	yes	—
34	C₈×Dic₁₁	—	2	2	no	—
35	C₈×Dic₁₁	—	2	2	no	—
36	C₈×Dic₁₁	—	2	2	no	—
37	C₈×Dic₁₁	C₈	1	4	yes	complemented
38	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
39	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
40	C₄×C₁₁⋊C₈	D₄	1	2	yes	—
41	C₄×C₁₁⋊C₈	Q₈	1	2	yes	—
42	#8807	—	11	1	no	maximal, Sylow
43	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₁	C₂×Q₈	1	1	yes	characteristic
46	C₂²×Dic₁₁	C₄⋊C₄	1	2	yes	—
47	C₂²×Dic₁₁	C₄⋊C₄	1	2	yes	—
48	C₂×C₈×C₁₁	—	2	1	no	—
49	C₂×C₈×C₁₁	—	2	1	no	—
50	C₂×C₈×C₁₁	—	4	2	no	—
51	C₂×C₈×C₁₁	—	4	4	no	—
52	C₂×C₈×C₁₁	C₄⋊C₄	1	2	yes	—
53	C₂×C₈×C₁₁	C₄⋊C₄	1	2	yes	—
54	C₂×C₈×C₁₁	C₂×C₈	1	2	yes	—
55	C₂×C₈×C₁₁	M₁₆	1	2	yes	—
56	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₁₁	C₂×Q₈	1	1	yes	characteristic
59	C₄²×C₁₁	C₂×Q₈	1	1	yes	characteristic
60	C₄×Dic₁₁	—	2	4	no	—
61	C₄×Dic₁₁	—	2	1	no	—
62	C₄×Dic₁₁	—	2	1	no	—
63	C₄×Dic₁₁	—	2	2	no	—
64	C₄×Dic₁₁	—	2	2	no	—
65	C₄×Dic₁₁	C₂×C₈	1	2	yes	—
66	C₄×Dic₁₁	M₁₆	1	2	yes	—
67	C₂×C₁₁⋊C₈	—	2	1	no	—
68	C₂×C₁₁⋊C₈	—	2	1	no	—
69	C₂×C₁₁⋊C₈	—	4	2	no	—
70	C₂×C₁₁⋊C₈	—	4	4	no	—
71	C₂×C₁₁⋊C₈	C₄⋊C₄	1	2	yes	—
72	C₂×C₁₁⋊C₈	C₄⋊C₄	1	2	yes	—
73	C₂×C₁₁⋊C₈	C₂×C₈	1	2	yes	—
74	C₂×C₁₁⋊C₈	M₁₆	1	2	yes	—
75	#1831	—	11	2	no	—
76	C₄²×C₈	—	11	1	no	—
77	C₂×#335	—	11	1	no	—
78	C₂×#335	D₁₁	1	1	yes	characteristic
79	C₂×#334	—	11	2	no	—
80	C₂³×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₁	—	2	2	no	—
82	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₁₁	#127	1	1	yes	characteristic
85	C₂×C₄×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
86	C₂×C₄×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
87	C₂×C₄×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
88	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
90	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
91	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
92	C₂×C₄×C₁₁	#135	1	2	yes	—
93	C₂×C₄×C₁₁	#135	1	2	yes	—
94	C₂×Dic₁₁	—	2	2	no	—
95	C₂×Dic₁₁	—	2	2	no	—
96	C₂×Dic₁₁	—	2	2	no	—
97	C₂×Dic₁₁	—	2	2	no	—
98	C₂×Dic₁₁	C₄⋊C₈	1	4	yes	—
99	C₂×Dic₁₁	#108	1	4	yes	—
100	C₈×C₁₁	—	2	2	no	—
101	C₈×C₁₁	—	8	2	no	—
102	C₈×C₁₁	C₄⋊C₈	1	4	yes	—
103	C₁₁⋊C₈	—	2	2	no	—
104	C₁₁⋊C₈	—	8	2	no	—
105	C₁₁⋊C₈	C₄⋊C₈	1	4	yes	—
106	C₂×C₄⋊C₈	—	11	2	no	—
107	C₂×C₄⋊C₈	C₂×D₁₁	1	2	yes	—
108	C₂×C₄×C₈	—	11	1	no	—
109	C₂×C₄×C₈	—	11	1	no	—
110	C₂×C₄×C₈	—	11	1	no	—
111	C₂×C₄×C₈	—	11	1	no	—
112	C₂×C₄×C₈	—	11	1	no	—
113	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
114	C₄³	—	11	1	no	—
115	#336	—	22	4	no	—
116	#335	—	2	2	no	—
117	#335	—	22	2	no	—
118	#334	—	22	4	no	—
119	C₂²×C₁₁	#384	1	1	yes	characteristic
120	C₂²×C₁₁	C₂×C₄⋊C₈	1	1	yes	characteristic
121	C₂²×C₁₁	#423	1	1	yes	characteristic
122	C₂²×C₁₁	#427	1	2	yes	—
123	C₂²×C₁₁	C₂×#108	1	2	yes	—
124	C₄×C₁₁	—	2	1	no	—
125	C₄×C₁₁	—	2	1	no	—
126	C₄×C₁₁	C₂×C₄⋊C₈	1	1	yes	characteristic
127	C₄×C₁₁	C₂×C₄⋊C₈	1	1	yes	characteristic
128	C₄×C₁₁	#423	1	1	yes	characteristic
129	C₄×C₁₁	#423	1	1	yes	characteristic
130	C₄×C₁₁	#495	1	1	yes	characteristic
131	C₄×C₁₁	#495	1	1	yes	characteristic
132	C₄×C₁₁	#499	1	2	yes	—
133	Dic₁₁	—	2	4	no	—
134	Dic₁₁	—	2	4	no	—
135	C₂²×C₈	—	2	2	no	—
136	C₂²×C₈	—	11	1	no	—
137	C₂²×C₈	—	11	1	no	—
138	C₂²×C₈	—	22	2	no	—
139	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
140	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
141	C₂×C₄²	—	11	1	no	—
142	C₂×C₄²	—	11	1	no	—
143	C₂×C₄²	—	11	2	no	—
144	C₂×C₄²	—	11	2	no	—
145	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
146	C₄⋊C₈	—	2	2	no	—
147	C₄⋊C₈	—	2	2	no	—
148	C₄⋊C₈	—	22	2	no	—
149	C₄⋊C₈	—	22	2	no	—
150	C₄×C₈	—	11	4	no	—
151	C₄×C₈	—	11	2	no	—
152	C₄×C₈	—	11	2	no	—
153	C₄×C₈	—	22	2	no	—
154	C₄×C₈	—	22	2	no	—
155	C₄×C₈	—	22	2	no	—
156	C₄×C₈	Dic₂₂	1	2	yes	—
157	C₄×C₈	C₂²⋊D₁₁	1	2	yes	—
158	C₂×C₁₁	#1564	1	2	yes	—
159	C₂×C₁₁	#1568	1	2	yes	—
160	C₂×C₁₁	#1842	1	1	yes	characteristic
161	C₂×C₁₁	#1846	1	1	yes	characteristic
162	C₂×C₁₁	#1848	1	1	yes	characteristic
163	C₂²×C₄	—	11	2	no	—
164	C₂²×C₄	—	11	2	no	—
165	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
166	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
167	C₂²×C₄	Q₈×D₁₁	1	1	yes	characteristic
168	C₂×C₈	—	2	1	no	—
169	C₂×C₈	—	2	1	no	—
170	C₂×C₈	—	4	4	no	—
171	C₂×C₈	—	4	2	no	—
172	C₂×C₈	—	11	2	no	—
173	C₂×C₈	—	11	2	no	—
174	C₂×C₈	—	11	2	no	—
175	C₂×C₈	—	11	2	no	—
176	C₂×C₈	—	22	1	no	—
177	C₂×C₈	—	22	1	no	—
178	C₂×C₈	—	44	4	no	—
179	C₂×C₈	—	44	2	no	—
180	C₂×C₈	C₈×D₁₁	1	2	yes	—
181	C₂×C₈	#4313	1	2	yes	—
182	C₂×C₈	Dic₁₁⋊C₄	1	2	yes	—
183	C₂×C₈	Dic₁₁⋊C₄	1	2	yes	—
184	C₄²	—	11	2	no	—
185	C₄²	—	11	2	no	—
186	C₄²	—	22	4	no	—
187	C₄²	—	22	1	no	—
188	C₄²	—	22	1	no	—
189	C₄²	—	22	2	no	—
190	C₄²	—	22	2	no	—
191	C₄²	C₂×Dic₂₂	1	1	yes	characteristic
192	C₄²	D₄×D₁₁	1	1	yes	characteristic
193	C₄²	Q₈×D₁₁	1	1	yes	characteristic
194	C₄²	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
195	C₁₁	#8807	1	1	yes	characteristic, complemented, Sylow
196	C₂³	C₄⋊C₄×D₁₁	1	1	yes	characteristic
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	11	4	no	—
199	C₂×C₄	—	11	4	no	—
200	C₂×C₄	—	22	2	no	—
201	C₂×C₄	—	22	2	no	—
202	C₂×C₄	—	22	2	no	—
203	C₂×C₄	—	22	2	no	—
204	C₂×C₄	#66941	1	1	yes	characteristic
205	C₂×C₄	#66941	1	1	yes	characteristic
206	C₂×C₄	C₄⋊C₄×D₁₁	1	1	yes	characteristic
207	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
208	C₂×C₄	C₂×#4313	1	1	yes	characteristic
209	C₂×C₄	M₁₆×D₁₁	1	1	yes	characteristic
210	C₂×C₄	M₁₆×D₁₁	1	1	yes	characteristic
211	C₂×C₄	#66965	1	2	yes	—
212	C₂×C₄	#66972	1	2	yes	—
213	C₂×C₄	C₂×Dic₁₁⋊C₄	1	1	yes	characteristic
214	C₂×C₄	#66993	1	1	yes	characteristic
215	C₂×C₄	#66997	1	1	yes	characteristic
216	C₈	—	2	2	no	—
217	C₈	—	8	2	no	—
218	C₈	—	11	4	no	—
219	C₈	—	22	2	no	—
220	C₈	—	88	2	no	—
221	C₈	Dic₁₁⋊C₈	1	4	yes	—
222	C₂²	D₁₁×C₄⋊C₈	1	1	yes	characteristic
223	C₂²	#10622450	1	1	yes	characteristic
224	C₂²	#10622493	1	2	yes	—
225	C₂²	D₁₁×#108	1	2	yes	—
226	C₂²	#10622589	1	1	yes	characteristic
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	22	4	no	—
230	C₄	—	22	4	no	—
231	C₄	D₁₁×C₄⋊C₈	1	1	yes	characteristic
232	C₄	#10622450	1	1	yes	characteristic
233	C₄	#10622491	1	2	yes	—
234	C₄	C₂×Dic₁₁⋊C₈	1	1	yes	characteristic
235	C₄	#10622731	1	1	yes	characteristic
236	C₄	#10622768	1	1	yes	characteristic
237	C₄	#10622804	1	1	yes	characteristic
238	C₂	#13067363	1	2	yes	—
239	C₂	#13067378	1	2	yes	—
240	C₂	#13068767	1	1	yes	characteristic
241	C₂	#13068779	1	1	yes	characteristic
242	C₂	#13068786	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₁₁
commutator	C₂×C₄×C₁₁	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₁
Fitting	C₂×C₁₁×#335	C₂
socle, prime radical	C₂³×C₁₁	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₁₁	C₂³
hypercenter	C₂×#335	D₁₁
hypocenter	C₁₁	#8807
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
37. C₈×Dic₁₁	C₈	semidirect
195. C₁₁	#8807	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#13068732	2
C₄×C₈×Dic₁₁	2
C₂×#10622096	2
C₂×#10622118	2
C₂×C₁₁×#335	2
#8807	11